# Properties

 Label 29.6.b.a Level $29$ Weight $6$ Character orbit 29.b Analytic conductor $4.651$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 29.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.65113077458$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 278 x^{10} + 28285 x^{8} + 1260472 x^{6} + 22944832 x^{4} + 140087936 x^{2} + 966400$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{14}\cdot 5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{4} q^{3} + ( -14 + \beta_{2} ) q^{4} + ( 4 - \beta_{6} ) q^{5} + ( 2 - \beta_{7} ) q^{6} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{7} + ( -10 \beta_{1} - \beta_{4} + \beta_{5} ) q^{8} + ( -131 - \beta_{3} + \beta_{6} - \beta_{10} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{4} q^{3} + ( -14 + \beta_{2} ) q^{4} + ( 4 - \beta_{6} ) q^{5} + ( 2 - \beta_{7} ) q^{6} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{7} + ( -10 \beta_{1} - \beta_{4} + \beta_{5} ) q^{8} + ( -131 - \beta_{3} + \beta_{6} - \beta_{10} ) q^{9} + ( 11 \beta_{1} + 4 \beta_{4} + \beta_{5} - \beta_{8} ) q^{10} + ( -21 \beta_{1} - 6 \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{11} + ( 13 \beta_{1} + 15 \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{11} ) q^{12} + ( 115 + 4 \beta_{2} - 3 \beta_{6} - 2 \beta_{7} + \beta_{10} ) q^{13} + ( 43 \beta_{1} - 5 \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{14} + ( -23 \beta_{1} - 11 \beta_{4} + 5 \beta_{5} + 3 \beta_{9} + \beta_{11} ) q^{15} + ( 28 + 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{6} + \beta_{7} + \beta_{10} ) q^{16} + ( 40 \beta_{1} + 12 \beta_{4} - \beta_{5} + \beta_{9} - 2 \beta_{11} ) q^{17} + ( -130 \beta_{1} - 22 \beta_{4} + \beta_{5} + 5 \beta_{8} ) q^{18} + ( 78 \beta_{1} - 7 \beta_{4} - 3 \beta_{5} - \beta_{8} + 3 \beta_{9} + \beta_{11} ) q^{19} + ( -376 - 6 \beta_{2} - 11 \beta_{3} + 7 \beta_{7} - 5 \beta_{10} ) q^{20} + ( -22 \beta_{1} - 42 \beta_{4} - 17 \beta_{5} + 6 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} ) q^{21} + ( 946 - 11 \beta_{2} + \beta_{3} - 20 \beta_{6} - 4 \beta_{7} + 5 \beta_{10} ) q^{22} + ( 483 - 6 \beta_{2} + 7 \beta_{3} + 8 \beta_{6} + 16 \beta_{7} + 4 \beta_{10} ) q^{23} + ( -538 - 33 \beta_{2} + 8 \beta_{3} + 24 \beta_{6} + 12 \beta_{7} - 4 \beta_{10} ) q^{24} + ( 1070 + 48 \beta_{2} - 5 \beta_{3} - 5 \beta_{6} - 2 \beta_{7} + 10 \beta_{10} ) q^{25} + ( 52 \beta_{1} + 122 \beta_{4} + 12 \beta_{5} - 8 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{26} + ( 156 \beta_{1} + 82 \beta_{4} - 3 \beta_{5} - 7 \beta_{8} + 3 \beta_{9} + \beta_{11} ) q^{27} + ( -2066 + 77 \beta_{2} - 33 \beta_{3} + 44 \beta_{6} - 23 \beta_{7} - 11 \beta_{10} ) q^{28} + ( 935 - 3 \beta_{1} - 12 \beta_{2} - 7 \beta_{3} - 26 \beta_{4} + 13 \beta_{5} + 20 \beta_{6} + 20 \beta_{7} - 4 \beta_{8} - 3 \beta_{9} + 5 \beta_{10} - \beta_{11} ) q^{29} + ( 1214 - 93 \beta_{2} + 29 \beta_{3} - 4 \beta_{6} - 40 \beta_{7} + 5 \beta_{10} ) q^{30} + ( -79 \beta_{1} + 72 \beta_{4} - 14 \beta_{5} + 13 \beta_{8} - 6 \beta_{9} - 2 \beta_{11} ) q^{31} + ( -423 \beta_{1} - 83 \beta_{4} + 20 \beta_{5} - \beta_{8} - 4 \beta_{9} + 3 \beta_{11} ) q^{32} + ( -1859 + 48 \beta_{2} - 35 \beta_{3} - 69 \beta_{6} + 42 \beta_{7} - 14 \beta_{10} ) q^{33} + ( -1852 + 49 \beta_{2} + 39 \beta_{3} - 40 \beta_{6} + 53 \beta_{7} - \beta_{10} ) q^{34} + ( 319 - 138 \beta_{2} + 5 \beta_{3} + 96 \beta_{6} - 52 \beta_{7} - 10 \beta_{10} ) q^{35} + ( 1866 - 165 \beta_{2} + 41 \beta_{3} - 104 \beta_{6} - 25 \beta_{7} - \beta_{10} ) q^{36} + ( 458 \beta_{1} - 120 \beta_{4} - 38 \beta_{5} - 8 \beta_{8} - 6 \beta_{9} - 2 \beta_{11} ) q^{37} + ( -3556 + 171 \beta_{2} - 9 \beta_{3} - 8 \beta_{6} - 51 \beta_{7} - 9 \beta_{10} ) q^{38} + ( -900 \beta_{1} - 320 \beta_{4} + 36 \beta_{5} + 9 \beta_{8} + 6 \beta_{9} - 3 \beta_{11} ) q^{39} + ( 191 \beta_{1} - 273 \beta_{4} + 40 \beta_{5} + \beta_{8} + 12 \beta_{9} + \beta_{11} ) q^{40} + ( 316 \beta_{1} + 244 \beta_{4} + 13 \beta_{5} - 4 \beta_{8} + 3 \beta_{9} + 10 \beta_{11} ) q^{41} + ( 812 + 297 \beta_{2} - 29 \beta_{3} - 176 \beta_{6} - 115 \beta_{7} + 19 \beta_{10} ) q^{42} + ( 241 \beta_{1} + 208 \beta_{4} + 29 \beta_{5} + 3 \beta_{8} + 21 \beta_{9} - 8 \beta_{11} ) q^{43} + ( 782 \beta_{1} + 185 \beta_{4} - 13 \beta_{5} - 8 \beta_{8} - 24 \beta_{9} - 8 \beta_{11} ) q^{44} + ( -4357 + 252 \beta_{2} + 54 \beta_{3} + 220 \beta_{6} + 42 \beta_{7} + 9 \beta_{10} ) q^{45} + ( 345 \beta_{1} - 712 \beta_{4} - 63 \beta_{5} + 5 \beta_{8} - 6 \beta_{9} + 19 \beta_{11} ) q^{46} + ( -378 \beta_{1} + 465 \beta_{4} - 15 \beta_{5} - 20 \beta_{8} - 11 \beta_{9} + 2 \beta_{11} ) q^{47} + ( 366 \beta_{1} - 243 \beta_{4} - 45 \beta_{5} + 8 \beta_{8} - 24 \beta_{9} - 8 \beta_{11} ) q^{48} + ( 8485 - 296 \beta_{2} - 54 \beta_{3} - 96 \beta_{6} + 96 \beta_{7} + 6 \beta_{10} ) q^{49} + ( -87 \beta_{1} + 276 \beta_{4} + 82 \beta_{5} - 32 \beta_{8} + 30 \beta_{9} - 17 \beta_{11} ) q^{50} + ( 4749 - 486 \beta_{2} - 9 \beta_{3} - 184 \beta_{6} - 4 \beta_{7} - 6 \beta_{10} ) q^{51} + ( 1230 - 49 \beta_{2} - 8 \beta_{3} + 116 \beta_{6} + 152 \beta_{7} - 4 \beta_{10} ) q^{52} + ( 2041 - 204 \beta_{2} + 40 \beta_{3} + 133 \beta_{6} + 34 \beta_{7} + 19 \beta_{10} ) q^{53} + ( -7346 + 267 \beta_{2} - 93 \beta_{3} + 160 \beta_{6} + 44 \beta_{7} - 45 \beta_{10} ) q^{54} + ( -2126 \beta_{1} + 558 \beta_{4} - 94 \beta_{5} + 59 \beta_{8} - 12 \beta_{9} + 29 \beta_{11} ) q^{55} + ( -2505 \beta_{1} + 674 \beta_{4} + 7 \beta_{5} + 55 \beta_{8} - 20 \beta_{9} - 13 \beta_{11} ) q^{56} + ( -2639 + 456 \beta_{2} + 2 \beta_{3} + 252 \beta_{6} - 106 \beta_{7} + 41 \beta_{10} ) q^{57} + ( 292 + 1039 \beta_{1} - 281 \beta_{2} - 31 \beta_{3} - 894 \beta_{4} - 46 \beta_{5} + 188 \beta_{6} + 43 \beta_{7} + 32 \beta_{8} + 24 \beta_{9} - 11 \beta_{10} + 8 \beta_{11} ) q^{58} + ( -517 - 570 \beta_{2} - 19 \beta_{3} + 80 \beta_{6} - 100 \beta_{7} - 34 \beta_{10} ) q^{59} + ( 3174 \beta_{1} + 1679 \beta_{4} + 69 \beta_{5} - 88 \beta_{8} + 48 \beta_{9} + 16 \beta_{11} ) q^{60} + ( -1228 \beta_{1} + 74 \beta_{4} - 41 \beta_{5} - 18 \beta_{8} + 33 \beta_{9} - 28 \beta_{11} ) q^{61} + ( 3218 + 102 \beta_{2} + 112 \beta_{3} - 372 \beta_{6} + 109 \beta_{7} + 64 \beta_{10} ) q^{62} + ( -11892 + 864 \beta_{2} - 44 \beta_{3} - 212 \beta_{6} + 120 \beta_{7} - 8 \beta_{10} ) q^{63} + ( 20702 - 757 \beta_{2} + 54 \beta_{3} + 320 \beta_{6} - 62 \beta_{7} + 46 \beta_{10} ) q^{64} + ( 14108 - 216 \beta_{2} - 33 \beta_{3} - 335 \beta_{6} - 60 \beta_{7} + 21 \beta_{10} ) q^{65} + ( -2776 \beta_{1} - 1956 \beta_{4} + 124 \beta_{5} + 50 \beta_{8} + 42 \beta_{9} + 21 \beta_{11} ) q^{66} + ( -2072 + 832 \beta_{2} + 40 \beta_{3} - 188 \beta_{6} - 8 \beta_{7} + 76 \beta_{10} ) q^{67} + ( -2799 \beta_{1} - 2094 \beta_{4} - 167 \beta_{5} - 23 \beta_{8} - 48 \beta_{9} + 29 \beta_{11} ) q^{68} + ( 5554 \beta_{1} - 978 \beta_{4} - 115 \beta_{5} - 34 \beta_{8} - 57 \beta_{9} - 10 \beta_{11} ) q^{69} + ( 3953 \beta_{1} + 1988 \beta_{4} - 155 \beta_{5} + 69 \beta_{8} - 30 \beta_{9} - 37 \beta_{11} ) q^{70} + ( -23706 + 420 \beta_{2} + 14 \beta_{3} - 68 \beta_{6} + 80 \beta_{7} - 136 \beta_{10} ) q^{71} + ( 2749 \beta_{1} + 950 \beta_{4} - 103 \beta_{5} - 7 \beta_{8} - 84 \beta_{9} + 17 \beta_{11} ) q^{72} + ( 3876 \beta_{1} - 8 \beta_{4} + 36 \beta_{5} + 64 \beta_{8} + 60 \beta_{9} + 20 \beta_{11} ) q^{73} + ( -21520 + 1182 \beta_{2} - 254 \beta_{3} + 120 \beta_{6} - 110 \beta_{7} - 86 \beta_{10} ) q^{74} + ( -1627 \beta_{1} - 2382 \beta_{4} + 298 \beta_{5} - 28 \beta_{8} - 63 \beta_{11} ) q^{75} + ( -5159 \beta_{1} + 1872 \beta_{4} + 203 \beta_{5} - 55 \beta_{8} + 96 \beta_{9} - 19 \beta_{11} ) q^{76} + ( 78 \beta_{1} - 1766 \beta_{4} + 307 \beta_{5} - 26 \beta_{8} + 65 \beta_{9} + 14 \beta_{11} ) q^{77} + ( 42718 - 1632 \beta_{2} + 342 \beta_{3} - 216 \beta_{6} - 215 \beta_{7} + 90 \beta_{10} ) q^{78} + ( -3372 \beta_{1} - 753 \beta_{4} - 129 \beta_{5} + 6 \beta_{8} - 117 \beta_{9} + 42 \beta_{11} ) q^{79} + ( -19446 - 711 \beta_{2} - 114 \beta_{3} - 54 \beta_{7} - 114 \beta_{10} ) q^{80} + ( -1990 + 96 \beta_{2} - 56 \beta_{3} + 268 \beta_{6} - 370 \beta_{7} - 89 \beta_{10} ) q^{81} + ( -14812 + 179 \beta_{2} - 147 \beta_{3} + 248 \beta_{6} + 19 \beta_{7} - 11 \beta_{10} ) q^{82} + ( 7131 - 90 \beta_{2} + 381 \beta_{3} - 180 \beta_{6} - 180 \beta_{7} + 30 \beta_{10} ) q^{83} + ( -5191 \beta_{1} + 4906 \beta_{4} + 265 \beta_{5} - 127 \beta_{8} - 99 \beta_{11} ) q^{84} + ( -1602 \beta_{1} + 2610 \beta_{4} - 63 \beta_{5} - 138 \beta_{8} + 27 \beta_{9} + 66 \beta_{11} ) q^{85} + ( -10606 - 273 \beta_{2} + 467 \beta_{3} - 316 \beta_{6} + 350 \beta_{7} + 47 \beta_{10} ) q^{86} + ( -11793 + 7051 \beta_{1} - 1086 \beta_{2} + 135 \beta_{3} - 874 \beta_{4} + 2 \beta_{5} - 104 \beta_{6} - 104 \beta_{7} - 101 \beta_{8} - 54 \beta_{9} - 84 \beta_{10} - 18 \beta_{11} ) q^{87} + ( -6796 + 474 \beta_{2} - 231 \beta_{3} - 276 \beta_{6} + 351 \beta_{7} + 99 \beta_{10} ) q^{88} + ( 3132 \beta_{1} + 2916 \beta_{4} - 13 \beta_{5} + 160 \beta_{8} - 115 \beta_{9} - 30 \beta_{11} ) q^{89} + ( -14117 \beta_{1} - 3034 \beta_{4} - 205 \beta_{5} + 181 \beta_{8} - 90 \beta_{9} + 87 \beta_{11} ) q^{90} + ( -15269 - 98 \beta_{2} - 15 \beta_{3} + 236 \beta_{6} - 596 \beta_{7} - 2 \beta_{10} ) q^{91} + ( -22 + 1527 \beta_{2} - 259 \beta_{3} + 164 \beta_{6} - 757 \beta_{7} + 95 \beta_{10} ) q^{92} + ( 31764 + 408 \beta_{2} - 302 \beta_{3} - 769 \beta_{6} + 436 \beta_{7} - 8 \beta_{10} ) q^{93} + ( 15922 - 77 \beta_{2} - 467 \beta_{3} + 656 \beta_{6} + 462 \beta_{7} - 135 \beta_{10} ) q^{94} + ( -515 \beta_{1} - 2873 \beta_{4} + 65 \beta_{5} - 69 \beta_{8} - 51 \beta_{9} - 80 \beta_{11} ) q^{95} + ( -34676 - 54 \beta_{2} + 121 \beta_{3} + 556 \beta_{6} + 355 \beta_{7} - 125 \beta_{10} ) q^{96} + ( -11848 \beta_{1} - 2052 \beta_{4} + \beta_{5} + 4 \beta_{8} + 111 \beta_{9} - 98 \beta_{11} ) q^{97} + ( 17181 \beta_{1} - 3604 \beta_{4} - 224 \beta_{5} + 36 \beta_{8} + 120 \beta_{9} + 36 \beta_{11} ) q^{98} + ( 11378 \beta_{1} + 4243 \beta_{4} + 457 \beta_{5} - 197 \beta_{8} + 111 \beta_{9} + 49 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 172 q^{4} + 46 q^{5} + 24 q^{6} + 20 q^{7} - 1574 q^{9} + O(q^{10})$$ $$12 q - 172 q^{4} + 46 q^{5} + 24 q^{6} + 20 q^{7} - 1574 q^{9} + 1362 q^{13} + 340 q^{16} - 4508 q^{20} + 11376 q^{22} + 5852 q^{23} - 6292 q^{24} + 12678 q^{25} - 25056 q^{28} + 11328 q^{29} + 14952 q^{30} - 22694 q^{33} - 22504 q^{34} + 4532 q^{35} + 22840 q^{36} - 43408 q^{38} + 8280 q^{42} - 52816 q^{45} + 102836 q^{49} + 58540 q^{51} + 15172 q^{52} + 25650 q^{53} - 89080 q^{54} - 32824 q^{57} + 4960 q^{58} - 3900 q^{59} + 37720 q^{62} - 146616 q^{63} + 252276 q^{64} + 169574 q^{65} - 28264 q^{67} - 286832 q^{71} - 263072 q^{74} + 519072 q^{78} - 230964 q^{80} - 24084 q^{81} - 178008 q^{82} + 85692 q^{83} - 126624 q^{86} - 137716 q^{87} - 83604 q^{88} - 182372 q^{91} - 5664 q^{92} + 377966 q^{93} + 192144 q^{94} - 415284 q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 278 x^{10} + 28285 x^{8} + 1260472 x^{6} + 22944832 x^{4} + 140087936 x^{2} + 966400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 46$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{10} - 18002 \nu^{8} - 3564589 \nu^{6} - 193044948 \nu^{4} - 2100465264 \nu^{2} + 3169589568$$$$)/37604096$$ $$\beta_{4}$$ $$=$$ $$($$$$-125 \nu^{11} - 35271 \nu^{9} - 3622703 \nu^{7} - 161356045 \nu^{5} - 2877987192 \nu^{3} - 17080930576 \nu$$$$)/56406144$$ $$\beta_{5}$$ $$=$$ $$($$$$-125 \nu^{11} - 35271 \nu^{9} - 3622703 \nu^{7} - 161356045 \nu^{5} - 2821581048 \nu^{3} - 12906875920 \nu$$$$)/56406144$$ $$\beta_{6}$$ $$=$$ $$($$$$-543 \nu^{10} - 114438 \nu^{8} - 7870715 \nu^{6} - 185889684 \nu^{4} - 714248592 \nu^{2} + 7095849024$$$$)/75208192$$ $$\beta_{7}$$ $$=$$ $$($$$$-521 \nu^{10} - 87078 \nu^{8} - 3797045 \nu^{6} - 9883192 \nu^{4} + 430061424 \nu^{2} + 233612288$$$$)/56406144$$ $$\beta_{8}$$ $$=$$ $$($$$$-4129 \nu^{11} - 1048734 \nu^{9} - 96066205 \nu^{7} - 3784789952 \nu^{5} - 59476865040 \nu^{3} - 302055473792 \nu$$$$)/ 225624576$$ $$\beta_{9}$$ $$=$$ $$($$$$-3967 \nu^{11} - 1087632 \nu^{9} - 109815991 \nu^{7} - 4927586774 \nu^{5} - 93368650752 \nu^{3} - 609397253600 \nu$$$$)/ 112812288$$ $$\beta_{10}$$ $$=$$ $$($$$$4237 \nu^{10} + 1022802 \nu^{8} + 86899681 \nu^{6} + 2985321308 \nu^{4} + 32933911152 \nu^{2} + 30413683520$$$$)/ 112812288$$ $$\beta_{11}$$ $$=$$ $$($$$$-7485 \nu^{11} - 2070898 \nu^{9} - 209723921 \nu^{7} - 9293509212 \nu^{5} - 167479043248 \nu^{3} - 1002999387456 \nu$$$$)/75208192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 46$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - \beta_{4} - 74 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{10} + \beta_{7} + 4 \beta_{6} + 3 \beta_{3} - 94 \beta_{2} + 3420$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{11} - 4 \beta_{9} - \beta_{8} - 108 \beta_{5} + 45 \beta_{4} + 5977 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-114 \beta_{10} - 222 \beta_{7} - 320 \beta_{6} - 426 \beta_{3} + 8139 \beta_{2} - 276642$$ $$\nu^{7}$$ $$=$$ $$-534 \beta_{11} + 624 \beta_{9} + 226 \beta_{8} + 10067 \beta_{5} + 2147 \beta_{4} - 494572 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$11423 \beta_{10} + 31751 \beta_{7} + 20044 \beta_{6} + 48149 \beta_{3} - 697028 \beta_{2} + 22902656$$ $$\nu^{9}$$ $$=$$ $$68477 \beta_{11} - 73452 \beta_{9} - 30623 \beta_{8} - 913598 \beta_{5} - 743283 \beta_{4} + 41324323 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-1097336 \beta_{10} - 3816044 \beta_{7} - 1093800 \beta_{6} - 4999668 \beta_{3} + 59790281 \beta_{2} - 1914098054$$ $$\nu^{11}$$ $$=$$ $$-7718376 \beta_{11} + 7804664 \beta_{9} + 3381832 \beta_{8} + 82417837 \beta_{5} + 111991603 \beta_{4} - 3475181502 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/29\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 − 9.47123i − 9.02264i − 8.60273i − 4.44887i − 3.61683i − 0.0831044i 0.0831044i 3.61683i 4.44887i 8.60273i 9.02264i 9.47123i
9.47123i 19.9099i −57.7042 −4.84615 −188.571 219.131 243.450i −153.403 45.8990i
28.2 9.02264i 26.5755i −49.4080 71.1177 239.781 131.830 157.067i −463.255 641.670i
28.3 8.60273i 4.65015i −42.0069 −58.8511 40.0039 −192.738 86.0865i 221.376 506.279i
28.4 4.44887i 21.5070i 12.2075 90.0922 −95.6821 −211.678 196.674i −219.553 400.809i
28.5 3.61683i 5.13082i 18.9185 15.9022 18.5573 69.7674 184.164i 216.675 57.5154i
28.6 0.0831044i 25.1364i 31.9931 −90.4149 −2.08895 −6.31225 5.31810i −388.840 7.51387i
28.7 0.0831044i 25.1364i 31.9931 −90.4149 −2.08895 −6.31225 5.31810i −388.840 7.51387i
28.8 3.61683i 5.13082i 18.9185 15.9022 18.5573 69.7674 184.164i 216.675 57.5154i
28.9 4.44887i 21.5070i 12.2075 90.0922 −95.6821 −211.678 196.674i −219.553 400.809i
28.10 8.60273i 4.65015i −42.0069 −58.8511 40.0039 −192.738 86.0865i 221.376 506.279i
28.11 9.02264i 26.5755i −49.4080 71.1177 239.781 131.830 157.067i −463.255 641.670i
28.12 9.47123i 19.9099i −57.7042 −4.84615 −188.571 219.131 243.450i −153.403 45.8990i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.6.b.a 12
3.b odd 2 1 261.6.c.b 12
4.b odd 2 1 464.6.e.c 12
29.b even 2 1 inner 29.6.b.a 12
29.c odd 4 2 841.6.a.d 12
87.d odd 2 1 261.6.c.b 12
116.d odd 2 1 464.6.e.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.b.a 12 1.a even 1 1 trivial
29.6.b.a 12 29.b even 2 1 inner
261.6.c.b 12 3.b odd 2 1
261.6.c.b 12 87.d odd 2 1
464.6.e.c 12 4.b odd 2 1
464.6.e.c 12 116.d odd 2 1
841.6.a.d 12 29.c odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(29, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$966400 + 140087936 T^{2} + 22944832 T^{4} + 1260472 T^{6} + 28285 T^{8} + 278 T^{10} + T^{12}$$
$3$ $$46577165867100 + 4281127461369 T^{2} + 112977325989 T^{4} + 715200530 T^{6} + 1884878 T^{8} + 2245 T^{10} + T^{12}$$
$5$ $$( -2627317458 - 384523443 T + 33953337 T^{2} + 235866 T^{3} - 12280 T^{4} - 23 T^{5} + T^{6} )^{2}$$
$7$ $$( -519034134784 - 73626194400 T + 1377655664 T^{2} + 1925088 T^{3} - 76080 T^{4} - 10 T^{5} + T^{6} )^{2}$$
$11$ $$77\!\cdots\!00$$$$+$$$$54\!\cdots\!09$$$$T^{2} +$$$$40\!\cdots\!93$$$$T^{4} + 66207806581653490 T^{6} + 398836040622 T^{8} + 1042869 T^{10} + T^{12}$$
$13$ $$( -1586247413057582 - 11115625280661 T + 39478948905 T^{2} + 236495534 T^{3} - 458892 T^{4} - 681 T^{5} + T^{6} )^{2}$$
$17$ $$52\!\cdots\!00$$$$+$$$$51\!\cdots\!76$$$$T^{2} +$$$$11\!\cdots\!44$$$$T^{4} +$$$$10\!\cdots\!40$$$$T^{6} + 49655547709152 T^{8} + 11376404 T^{10} + T^{12}$$
$19$ $$51\!\cdots\!00$$$$+$$$$98\!\cdots\!64$$$$T^{2} +$$$$62\!\cdots\!72$$$$T^{4} +$$$$10\!\cdots\!84$$$$T^{6} + 60421932252544 T^{8} + 13202492 T^{10} + T^{12}$$
$23$ $$( -31226156863382423808 - 2503955050471584 T + 56060704650096 T^{2} + 19526838816 T^{3} - 13642480 T^{4} - 2926 T^{5} + T^{6} )^{2}$$
$29$ $$74\!\cdots\!01$$$$-$$$$41\!\cdots\!72$$$$T +$$$$70\!\cdots\!78$$$$T^{2} +$$$$12\!\cdots\!36$$$$T^{3} -$$$$43\!\cdots\!73$$$$T^{4} -$$$$75\!\cdots\!68$$$$T^{5} +$$$$46\!\cdots\!36$$$$T^{6} - 3663693278490800832 T^{7} - 1033792375290073 T^{8} + 150493841664 T^{9} + 40051378 T^{10} - 11328 T^{11} + T^{12}$$
$31$ $$16\!\cdots\!00$$$$+$$$$15\!\cdots\!21$$$$T^{2} +$$$$54\!\cdots\!13$$$$T^{4} +$$$$88\!\cdots\!58$$$$T^{6} + 6381230863566862 T^{8} + 148696173 T^{10} + T^{12}$$
$37$ $$49\!\cdots\!00$$$$+$$$$13\!\cdots\!04$$$$T^{2} +$$$$11\!\cdots\!20$$$$T^{4} +$$$$37\!\cdots\!52$$$$T^{6} + 56603763382251520 T^{8} + 392424816 T^{10} + T^{12}$$
$41$ $$37\!\cdots\!00$$$$+$$$$19\!\cdots\!04$$$$T^{2} +$$$$28\!\cdots\!76$$$$T^{4} +$$$$17\!\cdots\!04$$$$T^{6} + 45044729846919648 T^{8} + 455380116 T^{10} + T^{12}$$
$43$ $$12\!\cdots\!00$$$$+$$$$30\!\cdots\!81$$$$T^{2} +$$$$22\!\cdots\!73$$$$T^{4} +$$$$74\!\cdots\!94$$$$T^{6} + 108261064070911726 T^{8} + 592865277 T^{10} + T^{12}$$
$47$ $$12\!\cdots\!00$$$$+$$$$51\!\cdots\!21$$$$T^{2} +$$$$28\!\cdots\!01$$$$T^{4} +$$$$53\!\cdots\!22$$$$T^{6} + 377271643969171342 T^{8} + 1055757077 T^{10} + T^{12}$$
$53$ $$( -$$$$28\!\cdots\!50$$$$-$$$$13\!\cdots\!45$$$$T + 45758547464456889 T^{2} + 3296230288398 T^{3} - 431047748 T^{4} - 12825 T^{5} + T^{6} )^{2}$$
$59$ $$( -$$$$28\!\cdots\!32$$$$-$$$$23\!\cdots\!20$$$$T + 1023607654738132464 T^{2} + 784970375712 T^{3} - 1999071800 T^{4} + 1950 T^{5} + T^{6} )^{2}$$
$61$ $$32\!\cdots\!00$$$$+$$$$34\!\cdots\!04$$$$T^{2} +$$$$85\!\cdots\!76$$$$T^{4} +$$$$78\!\cdots\!28$$$$T^{6} + 2678678398973202592 T^{8} + 3240969828 T^{10} + T^{12}$$
$67$ $$( -$$$$16\!\cdots\!04$$$$+$$$$13\!\cdots\!56$$$$T + 3744117875783401472 T^{2} - 33716967702016 T^{3} - 3853518880 T^{4} + 14132 T^{5} + T^{6} )^{2}$$
$71$ $$( -$$$$24\!\cdots\!00$$$$-$$$$41\!\cdots\!04$$$$T - 22069958550469963776 T^{2} - 371339833644288 T^{3} + 3128506976 T^{4} + 143416 T^{5} + T^{6} )^{2}$$
$73$ $$22\!\cdots\!00$$$$+$$$$21\!\cdots\!36$$$$T^{2} +$$$$66\!\cdots\!52$$$$T^{4} +$$$$90\!\cdots\!44$$$$T^{6} + 53792491762804178944 T^{8} + 13004997824 T^{10} + T^{12}$$
$79$ $$23\!\cdots\!00$$$$+$$$$10\!\cdots\!21$$$$T^{2} +$$$$18\!\cdots\!37$$$$T^{4} +$$$$16\!\cdots\!90$$$$T^{6} + 77737629590866141902 T^{8} + 16230097053 T^{10} + T^{12}$$
$83$ $$($$$$11\!\cdots\!28$$$$-$$$$10\!\cdots\!60$$$$T + 17614267256046353904 T^{2} + 469412363138400 T^{3} - 10722002904 T^{4} - 42846 T^{5} + T^{6} )^{2}$$
$89$ $$13\!\cdots\!00$$$$+$$$$41\!\cdots\!16$$$$T^{2} +$$$$27\!\cdots\!16$$$$T^{4} +$$$$69\!\cdots\!24$$$$T^{6} +$$$$82\!\cdots\!28$$$$T^{8} + 46426278804 T^{10} + T^{12}$$
$97$ $$21\!\cdots\!00$$$$+$$$$42\!\cdots\!56$$$$T^{2} +$$$$13\!\cdots\!64$$$$T^{4} +$$$$28\!\cdots\!12$$$$T^{6} +$$$$22\!\cdots\!16$$$$T^{8} + 78943793076 T^{10} + T^{12}$$